Question

given the function f(x)=x^2 + 2x, determine the slope of the tangent line of f at x = 8 using the limit shown below. you do not have to simplify your answer.

Explanation:

Step1: Recall the limit - definition of the derivative

The slope of the tangent line of a function $y = f(x)$ at $x=a$ is given by $m=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a = 8$ and $f(x)=x^{2}+2x$.

Step2: Find $f(8 + h)$ and $f(8)$

First, find $f(8 + h)$:

$$
LATEXBLOCK0
$$

Next, find $f(8)$:

$$ f(8)=8^{2}+2\times8=64 + 16=80 $$

Step3: Substitute into the limit - formula

$$
LATEXBLOCK1
$$

Answer:

$\lim_{h
ightarrow0}\frac{(8 + h)^{2}+2(8 + h)-(8^{2}+2\times8)}{h}=\lim_{h
ightarrow0}\frac{h^{2}+18h}{h}$